Chapter 9 Input Modeling

1
Chapter 9 Input Modeling

• Banks, Carson, Nelson Nicol
• Discrete-Event System Simulation

2
Purpose Overview

• Input models provide the driving force for a
  simulation model.
• The quality of the output is no better than the
  quality of inputs.
• In this chapter, we will discuss the 4 steps of
  input model development
• Collect data from the real system
• Identify a probability distribution to represent
  the input process
• Choose parameters for the distribution
• Evaluate the chosen distribution and parameters
  for goodness of fit.

3
Data Collection

• One of the biggest tasks in solving a real
  problem. GIGO garbage-in-garbage-out
• Suggestions that may enhance and facilitate data
  collection
• Plan ahead begin by a practice or pre-observing
  session, watch for unusual circumstances
• Analyze the data as it is being collected check
  adequacy
• Combine homogeneous data sets, e.g. successive
  time periods, during the same time period on
  successive days
• Be aware of data censoring the quantity is not
  observed in its entirety, danger of leaving out
  long process times
• Check for relationship between variables, e.g.
  build scatter diagram
• Check for autocorrelation
• Collect input data, not performance data

4
Identifying the Distribution

• Histograms
• Selecting families of distribution
• Parameter estimation
• Goodness-of-fit tests
• Fitting a non-stationary process

5
Histograms Identifying the distribution

• A frequency distribution or histogram is useful
  in determining the shape of a distribution
• The number of class intervals depends on
• The number of observations
• The dispersion of the data
• Suggested the square root of the sample size
• For continuous data
• Corresponds to the probability density function
  of a theoretical distribution
• For discrete data
• Corresponds to the probability mass function
• If few data points are available combine
  adjacent cells to eliminate the ragged appearance
  of the histogram

6
Histograms Identifying the distribution

• Vehicle Arrival Example of vehicles arriving
  at an intersection between 7 am and 705 am was
  monitored for 100 random workdays.
• There are ample data, so the histogram may have a
  cell for each possible value in the data range

Same data with different interval sizes
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Selecting the Family of Distributions
Identifying the distribution

• A family of distributions is selected based on
• The context of the input variable
• Shape of the histogram
• Frequently encountered distributions
• Easier to analyze exponential, normal and
  Poisson
• Harder to analyze beta, gamma and Weibull

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Selecting the Family of Distributions
Identifying the distribution

• Use the physical basis of the distribution as a
  guide, for example
• Binomial of successes in n trials
• Poisson of independent events that occur in a
  fixed amount of time or space
• Normal distn of a process that is the sum of a
  number of component processes
• Exponential time between independent events, or
  a process time that is memoryless
• Weibull time to failure for components
• Discrete or continuous uniform models complete
  uncertainty
• Triangular a process for which only the minimum,
  most likely, and maximum values are known
• Empirical resamples from the actual data
  collected

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Selecting the Family of Distributions
Identifying the distribution

• Remember the physical characteristics of the
  process
• Is the process naturally discrete or continuous
  valued?
• Is it bounded?
• No true distribution for any stochastic input
  process
• Goal obtain a good approximation

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Quantile-Quantile Plots Identifying the
distribution

• Q-Q plot is a useful tool for evaluating
  distribution fit
• If X is a random variable with cdf F, then the
  q-quantile of X is the g such that
• When F has an inverse, g F-1(q)
• Let xi, i 1,2, ., n be a sample of data from
  X and yj, j 1,2, , n be the observations in
  ascending order
• where j is the ranking or order number

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Quantile-Quantile Plots Identifying the
distribution

• The plot of yj versus F-1( (j-0.5)/n) is
• Approximately a straight line if F is a member of
  an appropriate family of distributions
• The line has slope 1 if F is a member of an
  appropriate family of distributions with
  appropriate parameter values

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Quantile-Quantile Plots Identifying the
distribution

• Example Check whether the door installation
  times follows a normal distribution.
• The observations are now ordered from smallest to
  largest
• yj are plotted versus F-1( (j-0.5)/n) where F has
  a normal distribution with the sample mean (99.99
  sec) and sample variance (0.28322 sec2)

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Quantile-Quantile Plots Identifying the
distribution

• Example (continued) Check whether the door
  installation times follow a normal distribution.

Straight line, supporting the hypothesis of a
normal distribution
Superimposed density function of the normal
distribution
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Quantile-Quantile Plots Identifying the
distribution

• Consider the following while evaluating the
  linearity of a q-q plot
• The observed values never fall exactly on a
  straight line
• The ordered values are ranked and hence not
  independent, unlikely for the points to be
  scattered about the line
• Variance of the extremes is higher than the
  middle. Linearity of the points in the middle of
  the plot is more important.
• Q-Q plot can also be used to check homogeneity
• Check whether a single distribution can represent
  both sample sets
• Plotting the order values of the two data samples
  against each other

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Parameter Estimation Identifying the
distribution

• Next step after selecting a family of
  distributions
• If observations in a sample of size n are X1, X2,
  , Xn (discrete or continuous), the sample mean
  and variance are
• If the data are discrete and have been grouped in
  a frequency distribution
• where fj is the observed frequency of value Xj

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Parameter Estimation Identifying the
distribution

• When raw data are unavailable (data are grouped
  into class intervals), the approximate sample
  mean and variance are
• where fj is the observed frequency of in the
  jth class interval
• mj is the midpoint of the jth
  interval, and c is the number of class intervals
• A parameter is an unknown constant, but an
  estimator is a statistic.

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Parameter Estimation Identifying the
distribution

• Vehicle Arrival Example (continued) Table in the
  histogram example on slide 6 (Table 9.1 in book)
  can be analyzed to obtain
• The sample mean and variance are
• The histogram suggests X to have a Possion
  distribution
• However, note that sample mean is not equal to
  sample variance.
• Reason each estimator is a random variable, is
  not perfect.

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Goodness-of-Fit Tests Identifying the
distribution

• Conduct hypothesis testing on input data
  distribution using
• Kolmogorov-Smirnov test
• Chi-square test
• No single correct distribution in a real
  application exists.
• If very little data are available, it is unlikely
  to reject any candidate distributions
• If a lot of data are available, it is likely to
  reject all candidate distributions

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Chi-Square test Goodness-of-Fit Tests

• Intuition comparing the histogram of the data to
  the shape of the candidate density or mass
  function
• Valid for large sample sizes when parameters are
  estimated by maximum likelihood
• By arranging the n observations into a set of k
  class intervals or cells, the test statistics is
• which approximately follows the chi-square
  distribution with k-s-1 degrees of freedom, where
  s of parameters of the hypothesized
  distribution estimated by the sample statistics.

Expected Frequency Ei npi where pi is the
theoretical prob. of the ith interval. Suggested
Minimum 5
Observed Frequency
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Chi-Square test Goodness-of-Fit Tests

• The hypothesis of a chi-square test is
• H0 The random variable, X, conforms to the
  distributional assumption with the parameter(s)
  given by the estimate(s).
• H1 The random variable X does not conform.
• If the distribution tested is discrete and
  combining adjacent cell is not required (so that
  Ei gt minimum requirement)
• Each value of the random variable should be a
  class interval, unless combining is necessary, and

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Chi-Square test Goodness-of-Fit Tests

• If the distribution tested is continuous
• where ai-1 and ai are the endpoints of the ith
  class interval
• and f(x) is the assumed pdf, F(x) is the assumed
  cdf.
• Recommended number of class intervals (k)
• Caution Different grouping of data (i.e., k) can
  affect the hypothesis testing result.

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Chi-Square test Goodness-of-Fit Tests

• Vehicle Arrival Example (continued)
• H0 the random variable is Poisson
  distributed.
• H1 the random variable is not Poisson
  distributed.
• Degree of freedom is k-s-1 7-1-1 5, hence,
  the hypothesis is rejected at the 0.05 level of
  significance.

Combined because of min Ei
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Kolmogorov-Smirnov Test Goodness-of-Fit
Tests

• Intuition formalize the idea behind examining a
  q-q plot
• Recall from Chapter 7.4.1
• The test compares the continuous cdf, F(x), of
  the hypothesized distribution with the empirical
  cdf, SN(x), of the N sample observations.
• Based on the maximum difference statistics
  (Tabulated in A.8)
• D max F(x) - SN(x)
• A more powerful test, particularly useful when
• Sample sizes are small,
• No parameters have been estimated from the data.
• When parameter estimates have been made
• Critical values in Table A.8 are biased, too
  large.
• More conservative, i.e., smaller Type I error
  than specified.

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p-Values and Best Fits Goodness-of-Fit
Tests

• p-value for the test statistics
• The significance level at which one would just
  reject H0 for the given test statistic value.
• A measure of fit, the larger the better
• Large p-value good fit
• Small p-value poor fit
• Vehicle Arrival Example (cont.)
• H0 data is Possion
• Test statistics , with 5
  degrees of freedom
• p-value 0.00004, meaning we would reject H0
  with 0.00004 significance level, hence Poisson is
  a poor fit.

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p-Values and Best Fits Goodness-of-Fit
Tests

• Many software use p-value as the ranking measure
  to automatically determine the best fit.
  Things to be cautious about
• Software may not know about the physical basis of
  the data, distribution families it suggests may
  be inappropriate.
• Close conformance to the data does not always
  lead to the most appropriate input model.
• p-value does not say much about where the lack of
  fit occurs
• Recommended always inspect the automatic
  selection using graphical methods.

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Fitting a Non-stationary Poisson Process

• Fitting a NSPP to arrival data is difficult,
  possible approaches
• Fit a very flexible model with lots of parameters
  or
• Approximate constant arrival rate over some basic
  interval of time, but vary it from time interval
  to time interval.
• Suppose we need to model arrivals over time
  0,T, our approach is the most appropriate when
  we can
• Observe the time period repeatedly and
• Count arrivals / record arrival times.

Our focus
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Fitting a Non-stationary Poisson Process

• The estimated arrival rate during the ith time
  period is
• where n of observation periods, Dt time
  interval length
• Cij of arrivals during the ith time interval
  on the jth observation period
• Example Divide a 10-hour business day 8am,6pm
  into equal intervals k 20 whose length Dt ½,
  and observe over n 3 days

For instance, 1/3(0.5)(232632) 54
arrivals/hour
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Selecting Model without Data

• If data is not available, some possible sources
  to obtain information about the process are
• Engineering data often product or process has
  performance ratings provided by the manufacturer
  or company rules specify time or production
  standards.
• Expert option people who are experienced with
  the process or similar processes, often, they can
  provide optimistic, pessimistic and most-likely
  times, and they may know the variability as well.
• Physical or conventional limitations physical
  limits on performance, limits or bounds that
  narrow the range of the input process.
• The nature of the process.
• The uniform, triangular, and beta distributions
  are often used as input models.

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Selecting Model without Data

• Example Production planning simulation.
• Input of sales volume of various products is
  required, salesperson of product XYZ says that
• No fewer than 1,000 units and no more than 5,000
  units will be sold.
• Given her experience, she believes there is a 90
  chance of selling more than 2,000 units, a 25
  chance of selling more than 2,500 units, and only
  a 1 chance of selling more than 4,500 units.
• Translating these information into a cumulative
  probability of being less than or equal to those
  goals for simulation input

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Multivariate and Time-Series Input Models

• Multivariate
• For example, lead time and annual demand for an
  inventory model, increase in demand results in
  lead time increase, hence variables are
  dependent.
• Time-series
• For example, time between arrivals of orders to
  buy and sell stocks, buy and sell orders tend to
  arrive in bursts, hence, times between arrivals
  are dependent.

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Covariance and Correlation Multivariate/Tim
e Series

• Consider the model that describes relationship
  between X1 and X2
• b 0, X1 and X2 are statistically independent
• b gt 0, X1 and X2 tend to be above or below their
  means together
• b lt 0, X1 and X2 tend to be on opposite sides of
  their means
• Covariance between X1 and X2
• 0, 0
• where cov(X1, X2) lt 0, then b lt 0
• gt 0, gt 0

e is a random variable with mean 0 and is
independent of X2
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Covariance and Correlation Multivariate/Tim
e Series

• Correlation between X1 and X2 (values between -1
  and 1)
• 0, 0
• where corr(X1, X2) lt 0, then b lt 0
• gt 0, gt 0
• The closer r is to -1 or 1, the stronger the
  linear relationship is between X1 and X2.

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Covariance and Correlation Multivariate/Tim
e Series

• A time series is a sequence of random variables
  X1, X2, X3, , that are identically distributed
  (same mean and variance) but dependent.
• cov(Xt, Xth) is the lag-h autocovariance
• corr(Xt, Xth) is the lag-h autocorrelation
• If the autocovariance value depends only on h and
  not on t, the time series is covariance stationary

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Multivariate Input Models Multivariate/Time
Series

• If X1 and X2 are normally distributed, dependence
  between them can be modeled by the bivariate
  normal distribution with m1, m2, s12, s22 and
  correlation r
• To Estimate m1, m2, s12, s22, see Parameter
  Estimation (slide 15- 17, Section 9.3.2 in book)
• To Estimate r, suppose we have n independent and
  identically distributed pairs (X11, X21), (X12,
  X22), (X1n, X2n), then

Sample deviation
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Time-Series Input Models Multivariate/Time
Series

• If X1, X2, X3, is a sequence of identically
  distributed, but dependent and covariance-stationa
  ry random variables, then we can represent the
  process as follows
• Autoregressive order-1 model, AR(1)
• Exponential autoregressive order-1 model, EAR(1)
• Both have the characteristics that
• Lag-h autocorrelation decreases geometrically as
  the lag increases, hence, observations far apart
  in time are nearly independent

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AR(1) Time-Series Input Models Multivariate
/Time Series

• Consider the time-series model
• If X1 is chosen appropriately, then
• X1, X2, are normally distributed with mean m,
  and variance s2/(1-f2)
• Autocorrelation rh fh
• To estimate f, m, se2

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EAR(1) Time-Series Input Models Multivariat
e/Time Series

• Consider the time-series model
• If X1 is chosen appropriately, then
• X1, X2, are exponentially distributed with mean
  1/l
• Autocorrelation rh fh , and only positive
  correlation is allowed.
• To estimate f, l

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Summary

• In this chapter, we described the 4 steps in
  developing input data models
• Collecting the raw data
• Identifying the underlying statistical
  distribution
• Estimating the parameters
• Testing for goodness of fit